Most of my early articles, Geometry for Beginners, are about how to find a region of different polygons. For triangles, we learned that the formula A = 1/2 bh applications for all triangles. With right triangles, since the legs are perpendicular, one foot can be considered the base, and the other foot can be considered height. Unfortunately, the search for an area of irregular triangles is problematic, since neither side gives us the height that the formula requires. To find an area of irregular triangles, you need special skills.
Since the formula for the area of the triangle A = 1/2 bh applies to all triangles, we need to know the base and height of the triangle in question. Either side can be considered the base of a triangle, but solving the “no height” problem should be solved by creating a line segment that measures height. We do this by “dropping the perpendicular”, called height, to the base from the top opposite it.
Note: This height May beyond the limits of the triangle, so it may be necessary to expand the line containing the base. If the height falls inside a triangle, it divides the original triangle into two smaller triangles, which may or may not be the same size and shape. He also divides the base into two parts.
Let's try a practice situation. (Draw this image as I describe it.) Draw a base and name it 15 inches. Draw the left side - leaving it to the right - and make it shorter than the base and mark it 10 inches. Draw what should now be the third (right) side of the triangle. To find the area of the triangle, we need the height; therefore, we discard the perpendicular from the vertex vertex to the 15-inch base. Suppose that the two resulting parts of the base measure are 6 inches to the left and 9 inches to the right. (In fact, these measures depend on the angles of the triangle, so we simply use these numbers to make a point.)
We added perpendicular to this triangle so that we can have height. Do we know how long this happens? From the information provided directly, the answer is "is". NO. “So, the next question: can we understand what height is from what we know? Until now, we simply know how to find the missing side if we have a regular triangle with two known sides. Do we have it? In fact, we do! The added height created two right-angled triangles, and on the left, a hypotenuse of 10 inches (left side of the original triangle) and a base of 6 inches (left side of the base).
Now we have two choices about how to find the height: (1) check if we have a Pythagorean triple, or (2) use the Pythagorean theorem. I always suggest first looking for Pythagorean triples. Our two sides are 6 and 10, which have a total factor of 2, which leads to sides 3 and 5. This should make us think about 3, 4, 5 Pythagorean Triple. Thus, this leads us to 6, 8, 10 Pythagorean triple and a height of 8 inches. Is that logical? While the hypotenuse is longer, this is logical. So, h = 8 inches.
If you do not know your Pythagorean triples ... (1) go back and read my article about the Pythagorean paths and memorize them, and (2) find the missing side using the Pythagorean theorem. (This is also the best way to check the answer above. You never want to check the answer by repeating what you just did. Theorem c ^ 2 = a ^ 2 + b ^ 2 becomes 10 ^ 2 = 6 ^ 2 + b ^ 2 or 100 = 36 + b ^ 2 or 64 = b ^ 2 or b = +8 or -8 Since the length can be negative, b = 8, which is checked by our Pythagorean triple response.
Finally , we are ready to calculate the area, but ... ATTENTION! ATTENTION! ATTENTION! ERROR ABOUT ORIGIN! We have just finished working with a small triangle on the left to find the height, and many students get confused and find the area of this small triangle; BUT, our task is to find a region of a large initial triangle. Always re-read the original instructions after completing any one part. Then find the appropriate values for the task. Now A = 1/2 bh becomes A = (1/2) (15) (8) or A = 15 (4) = 60 square meters. Inches. (Remember that square is measured by squares.)
To read a point made earlier, it is best to check the answer in a different way than the method of finding the answer. For this problem, an alternative way to check our answer would be to find areas of two small triangles and add them together to find out if the total number we discovered for the area earlier. For the left triangle A = 1/2 (6) (8) or A = 24 square meters. Inches, and for the right triangle A = 1/2 (9) (8) or A = 36 square meters. Inches. the amount is 24 + 36 or 60 square meters. inches - exactly what we got before! YEA!
